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Maxwell Equations INEL 4151 ch 9 Sandra Cruz-Pol, Ph. D. ECE UPRM Mayag ü ez, PR

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Cruz-Pol, Electromagnetics UPRM Electricity => Magnetism In 1820 Oersted discovered that a steady current produces a magnetic field while teaching a physics class. In 1820 Oersted discovered that a steady current produces a magnetic field while teaching a physics class.

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Cruz-Pol, Electromagnetics UPRM Would magnetism would produce electricity? Eleven years later, and at the same time, Mike Faraday in London and Joe Henry in New York discovered that a time-varying magnetic field would produce an electric current! Eleven years later, and at the same time, Mike Faraday in London and Joe Henry in New York discovered that a time-varying magnetic field would produce an electric current! Faradays Law:

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Lens Law = (-) If N=1 (1 loop) If N=1 (1 loop) The time change The time change can refer to B or S Cruz-Pol, Electromagnetics UPRM

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Two cases of B changes B changes S (area) changes S (area) changes Cruz-Pol, Electromagnetics UPRM Stokes theo.

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3 cases: Stationary loop in time-varying B field Stationary loop in time-varying B field Moving loop in static B field Moving loop in static B field Moving loop in time-varying B field Moving loop in time-varying B field Cruz-Pol, Electromagnetics UPRM

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PE 9.1 If B= Cruz-Pol, Electromagnetics UPRM

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Example 9.3 Circuit with m 2 cross-section, radius =10cm, i 1 (t)=3 sin t [A], 50 Hz, N 1 =200 turns, N 2 = 100, =500 0, find V emf on both coils Find reluctance and use Faradays Law Find reluctance and use Faradays Law Cruz-Pol, Electromagnetics UPRM

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Example PE 9.3 A magnetic core of uniform cross-section 4 cm 2 is connected to a 120V, 60Hz generator. Calculate the induced emf V 2 in the secondary coil. N 1 = 500, N 2 =300 Use Faradays Law Use Faradays Law Answer; 72 cos(120 t) V

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Cruz-Pol, Electromagnetics UPRM Some terms E = electric field intensity [V/m] E = electric field intensity [V/m] D = electric field density [VF/m2] D = electric field density [VF/m2] H = magnetic field intensity, [A/m] H = magnetic field intensity, [A/m] B = magnetic field density, [Teslas] B = magnetic field density, [Teslas]

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Take a look at a capacitor dielectric is an electrical insulator dielectric is an electrical insulator d.c., C = open circuits d.c., C = open circuits Whats the current in an open circuit? Whats the current in an open circuit?

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Cruz-Pol, Electromagnetics UPRM Maxwells Eqs. the equation of continuity the equation of continuity Maxwell added the term to Amperes Law so that it not only works for static conditions but also for time-varying situations. Maxwell added the term to Amperes Law so that it not only works for static conditions but also for time-varying situations. This added term is called the displacement current density, J d, while J=J c is the conduction current. This added term is called the displacement current density, J d, while J=J c is the conduction current.

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For static fields we had: But the divergence of the curl of ANY vector 0 But the divergence of the curl of ANY vector 0 The continuity of the current requires: We need to define: Now when we take the divergence of curl Therefore: Cruz-Pol, Electromagnetics, UPRM This doesnt work for time-varying fields!

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Cruz-Pol, Electromagnetics UPRM Maxwell put them together If we didnt have the displacement current, J d, then Amperes Law: I S2S2 S1S1 L At low frequencies J>>J d, but at radio frequencies both terms are comparable in magnitude.

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For dynamic fields we had: For dynamic fields we had:Therefore: Substituting in curl of H This is Maxwells equation for Amperes Law Cruz-Pol, Electromagnetics, UPRM

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Cruz-Pol, Electromagnetics UPRM Maxwell Equations in General Form Differential form Integral Form Gausss Law for E field. Gausss Law for H field. Nonexistence of monopole Faradays Law Amperes Circuit Law

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Cruz-Pol, Electromagnetics UPRM Electromagnetics This is the principle of motors, hydro-electric generators and transformers operation. This is the principle of motors, hydro-electric generators and transformers operation. *Mention some examples of em waves This is what Oersted discovered accidentally:

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Cruz-Pol, Electromagnetics UPRM Phasors & complex #s Working with harmonic fields is easier, but requires knowledge of phasor, lets review complex numbers and complex numbers and phasors phasors

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Cruz-Pol, Electromagnetics UPRM Maxwell Equations for Harmonic fields Differential form* Gausss Law for E field. Gausss Law for H field. No monopole Faradays Law Amperes Circuit Law * (substituting and )

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Earth Magnetic Field Declination from 1590 to 1990 Cruz-Pol, Electromagnetics UPRM

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News Jan 2011 The Earth's magnetic north pole is slowly heading toward Russia, according to scientists, but one of the places being affected by this is Tampa International Airport. Airport officials closed its main runway this week until Jan. 13 to adjust the taxiway signs accounting for the magnetic pole shift, Tampa Bay Online reports. The Earth's magnetic north pole is slowly heading toward Russia, according to scientists, but one of the places being affected by this is Tampa International Airport. Airport officials closed its main runway this week until Jan. 13 to adjust the taxiway signs accounting for the magnetic pole shift, Tampa Bay Online reports.Tampa Bay Online reportsTampa Bay Online reports

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